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Multi angle

H 1 Criteria for equivalence between power series and polynomials

Auteurs : Kucharz, Wojciech (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Let $A$ be the ring of formal power series in $n$ variables over a field $K$ of characteristic zero. Two power series $f$ and $g$ in $A$ are said to be equivalent if there exists a $K$-automorphism of $A$ transforming $f$ into $g$. In my talk I will review criteria for a power series to be equivalent to a power series which is a polynomial in at least some of the variables. For example, each power series in $A$ is equivalent to a polynomial in two variables whose coefficients are power series in $n - 2$ variables. In particular, each power series in two variables over $K$ is equivalent to a polynomial with coefficients in $K$. Similar results are valid for convergent power series, assuming that the field $K$ is endowed with an absolute value and is complete. In the special case of convergent power series over the field of real numbers some weaker notions of equivalence will be also considered. I will report on works of several mathematicians giving simple proofs. Some open problems will be included.

singularities - power series

Codes MSC :
14B05 - Singularities
32B05 - Analytic algebras and generalizations, preparation theorems
58C25 - Differentiable maps

 Informations sur la Vidéo Langue : Anglais Date de publication : 17/02/15 Date de captation : 05/02/15 Collection : Research talks ; Algebraic and Complex Geometry Format : quicktime ; audio/x-aac Durée : 01:02:30 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-02-05_Kucharz.mp4 Informations sur la rencontre Nom de la rencontre : Applications of Artin approximation in singularity theory / Applications de l'approximation de Artin en théorie des singularitésOrganisateurs de la rencontre : Hauser, Herwig ; Rond, GuillaumeDates : 02/02/15 - 06/02/15 Année de la rencontre : 2015 URL Congrès : https://conferences.cirm-math.fr/1474.htmlCitation Data DOI : 10.24350/CIRM.V.18693103 Cite this video as: Kucharz, Wojciech (2015). Criteria for equivalence between power series and polynomials. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18693103 URI : http://dx.doi.org/10.24350/CIRM.V.18693103

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